Deflation and Balancing Preconditioners for Krylov Subspace Methods Applied to Nonsymmetric Matrices
Deflation and Balancing Preconditioners for Krylov Subspace Methods Applied to Nonsymmetric Matrices
复制标题
应用于非对称矩阵的 Krylov 子空间方法的紧缩和平衡预条件子
DOI：
10.1137/060678257
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
R. Nabben
中科院分区：
文献类型：
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作者：
Y. Erlangga;R. Nabben
在相当长的一段时间里，通货紧缩预条件因子被提出并用于加速Krylov子空间方法的收敛。对于对称正定线性系统，分析了与通缩相结合的共轭梯度法的收敛性，并与其他预条件方法进行了比较[R.Nabben和C.Vuik，SIAM J.Sci]。计算机，27(2006)，第1742-1759页]。本文将GMRES迭代下的收敛分析推广到非对称线性系统，并与抽象的非对称平衡预条件算子进行了比较。我们能够证明许多关于对称正定矩阵的结果推广到任意非对称矩阵。首先，我们证明了预条件系统的谱是相似的。此外，我们还证明了在一定条件下，GMRES结合通货紧缩产生的残差的2-范数永远不会大于GMRES结合抽象平衡预条件子产生的残差的2-范数。对对流扩散方程有限体积离散化产生的非对称线性方程组进行了数值实验，数值结果证实了我们的理论结果。
For quite some time, the deflation preconditioner has been proposed and used to accelerate the convergence of Krylov subspace methods. For symmetric positive definite linear systems, the convergence of conjugate gradient methods combined with deflation has been analyzed and compared with other preconditioners, e.g., with the abstract balancing preconditioner [R. Nabben and C. Vuik, SIAM J. Sci. Comput., 27 (2006), pp. 1742-1759]. In this paper, we extend the convergence analysis to nonsymmetric linear systems in the context of GMRES iteration and compare it with the abstract nonsymmetric balancing preconditioner. We are able to show that many results for symmetric positive definite matrices carry over to arbitrary nonsymmetric matrices. First we establish that the spectra of the preconditioned systems are similar. Moreover, we show that under certain conditions, the 2-norm of residuals produced by GMRES combined with deflation is never larger than the 2-norm of residuals produced by GMRES combined with the abstract balancing preconditioner. Numerical experiments are done to nonsymmetric linear systems arising from a finite volume discretization of the convection-diffusion equation, and the numerical results confirm our theoretical results.